This group comes equipped with a representation already, on the vector space itself! Just use the identity homomorphism We often call this the “standard” or “defining” representation. In fact, it’s easy to forget that it’s a representation at all. But it is.

As with any other group, we have dual representations. That is, we immediately get an action of on . And we’ve seen it already! When we talked about the coevaluation on vector spaces we worked out how a change of basis affects linear functionals. What we found is that if is our action on , then the action on is by the transpose — the dual — of . And this is exactly the dual representation.

Also, as with any other group, we have tensor representations — actions on the tensor power for any number of factors of . How does this work? Well, every vector in is a linear combination of vectors of the form , where each . And we know how to act on these: just act on each tensorand separately. That is,

Actually, I’m going to mention it tomorrow. But I’m not going to actually prove it, or use it as such. What I am going to do is introduce a particular representation that many readers may already be familiar with, but without picking a basis!

[…] tomorrow I want to take last Friday’s symmetrizer and antisymmetrizer and apply them to the tensor representations of , which we know also carry symmetric group representations. Specifically, the th tensor power […]

[…] project by considering antisymmetric tensors today. Remember that we’re starting with a tensor representation of on the tensor power , which also carries an action of the symmetric group by permuting the […]

[…] vector space. But remember that this isn’t just a vector space. The tensor power is both a representation of and a representation of , which actions commute with each other. Our antisymmetric tensors are […]

[…] each degree. And if is invertible, so must be its image under each functor. These give exactly the tensor, symmetric, and antisymmetric representations of the group , if we consider how these functors act […]

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.